The Mandarin Particle Dou:A Pre-Exhaustiﬁcation Exhaustiﬁer

EISS 11 1 The Mandarin Particle dou:A Pre-exhaustiﬁcation Exhaustiﬁer

Yimei Xiang

Abstract This paper provides a uniform semantics to capture various functions of Mandarin particle dou, including the quantifier-distributor use, FCI-licenser use, and scalar marker use. I argue that dou is a presuppositional exhaustifier that operates on sub-alternatives and has a pre-exhaustification effect.

Keywords dou · Exhaustiﬁcation · Quantiﬁcation · Free choice · Scalar · Alternative Semantics

Y. Xiang, Harvard University, http://scholar.harvard.edu/yxiang

In Christopher Piñón (ed.), Empirical Issues in Syntax and Semantics 11, 00–00. Paris: Colloque01 de Syntaxe et Sémantique à Paris (CSSP). http://www.cssp.cnrs.fr/eiss11/ © 2016 Yimei Xiang / 1 Introduction The Mandarin particle dou has various uses. Descriptively speaking, it can be used as a universal quantifier-distributor,03 a free choice item (FCI)- licenser, a scalar marker,Draft and so on. First, in a basic declarative sentence, the particle dou, similar to En- glish all, is associated with a preceding/ nominal expression and univer- sally quantifies and distributes over the subparts of this expression, as exemplified in (1). Here and throughout the paper, I use a [square bracket] to mark the item associated with dou.

(1) a. [Tamen] dou dao -le. they DOU arrive -ASP ‘They2016 all arrived.’ b. [Tamen] dou ba naxie wenti da dui -le. they DOU BA those question answer correct -ASP ‘They all correctly answered these questions.’ c. Tamen ba [naxie wenti] dou da dui -le. they BA those question DOU answer correct -ASP ‘They correctly answered all of these questions.’ 2 Y. Xiang

Moreover, under the quantiﬁer-distributor use, dou brings up three more semantic consequences in addition to universal quantiﬁcation, namely a “maximality requirement”, a “distributivity requirement”, and a “plurality requirement”. The “maximality requirement” means that dou forces the predicate denoted by the remnant VP to predicate on the maximal element in the extension of the associated item (Xiang 2008). For instance, imagine that a large group of children, with one or two exceptions, went to the park. Then (2) can be judged as true only when dou is absent.

(2) [Haizimen] (#dou) qu -le gongyuan. children DOU go -PERF park ‘The children (#all) went to the park.’

The “distributivity requirement” says that if a sentence admits both collective and atomic/nonatomic distributive readings, applying dou over this sentence blocks the collective reading (Lin 1998). For instance, (3a) is infelicitous if John and Mary married each other, and (3b) is infelicitous if all the considered individuals only participated in one house-buying event.

(3) a. [Yuehan he Mali] dou jiehun -le. John and Mary DOU get-married -ASP ‘John and Mary each got married.’ b. [Tamen] dou mai -le fangzi. they DOU buy -PERF house ‘They all bought houses.’ (# collective)

The “plurality requirement” says that the item associated with dou must take a non-atomic interpretation. If the prejacent sentence of dou has no overt non-atomic term, dou needs to be associated with a covert non- atomic item. For example in (4), since the spelled-out part of prejacent sentence has no non-singular term, dou is associated with a covert term such as zhe-ji-ci ‘these times’.

(4) Yuehan [(zhe-ji-ci)] dou qu de Beijing. John this-several-time DOU go DE Beijing ‘For all of these times, the place that John went to was Beijing.’ The Mandarin Particle dou: A Pre-exhaustiﬁcation Exhaustiﬁer 3

Second, as is well-known, dou can license a pre-verbal wh-item as a universal FCI, as exempliﬁed in (5). Moreover, I observe that dou in company with a possibility modal can license the universal FC use of a pre-verbal disjunction, as shown in (6). In particular, if the possibility modal keyi ‘can’ is dropped or replaced with a necessity modal bixu ‘must’, the use of dou makes the sentence ungrammatical.

(5) a. [Shui] (dou) he -guo jiu. who DOU drink -EXP alcohol ‘Anyone/everyone has had alcohol.’ b. [Na-ge nanhai] *(dou) he -guo hejiu. which-CL boy DOU drink -EXP alcohol ‘Any/Every boy has had alcohol.’ (6) a. [Yuehan huozhe Mali] (dou) keyi jiao hanyu. John or Mary DOU can teach Chinese Without dou: ‘Either John or Mary can teach Chinese.’ With dou: ‘Both John and Mary can teach Chinese.’ b. [Yuehan huozhe Mali] (*dou) jiao hanyu. John or Mary DOU teach Chinese c. [Yuehan huozhe Mali] (*dou) bixu jiao hanyu. John or Mary DOU must teach Chinese

Third, when associated with a scalar item, dou implies that the prejacent proposition ranks relatively high in the considered scale. When dou takes this use, its associated item can stay in-situ but must be focus- marked. Like in (7a), dou is associated with the numeral phrase wu dian ‘ﬁve o’clock’, and the alternatives are ranked in chronological order.

(7) a. Dou [WUF -dian] -le. DOU five-o’clock -ASP ‘It is five o’clock.’ Being five o’clock is a bit late. b. Ta dou lai -guo zher [LIANGF -ci] -le. he DOU come -EXP here two-time -ASP. ‘He has been here twice.’ Being here twice is a lot.

The [lian Foc dou ...] construction is a special case where dou functions as a scalar marker. A sentence taking a [lian Foc dou ...] construction has 4 Y. Xiang an ‘even’-like interpretation; it implicates that the prejacent proposition is less likely to be true than (most of) the contextually relevant alternatives.

(8) (Lian) [duizhang]F dou chi dao -le. LIAN team-leader DOU late arrive -ASP ‘Even [the team leader]F arrived late.’

In particular, ‘one-CL-NP’ can be licensed as a minimizer at the focus position of the [lian Foc dou NEG ...] construction, as shown in (9a). Notice that the post-dou negation is not always needed, as shown in (9b).

(9) a. Yuehan (lian) [YIF -ge ren] *(dou) *(mei) qing. John LIAN one-CL person DOU NEG invite ‘John didn’t invite even one person.’ b. Yuehan (lian) [YIF -fen qian] *(dou) (mei) yao. John LIAN one-cent money DOU NEG request Without negation: ‘John doesn’t want any money.’ With negation: ‘Even if it is just one cent, John wants it.’

In case that a sentence has multiple items that are eligible to be associated with dou, the function of dou and the association relation can be disambiguated by stress. Compare, in (10a), the prejacent of dou has no stressed item, dou functions as a quantiﬁer and is associated with the preceding plural term tamen ‘they’; while in (10b) and (10c), dou functions as a scalar marker and is associated with the stressed item.

(10) a. [Tamen] DOU/dou lai -guo liang-ci -le. they DOU/DOU come -EXP two-time -ASP ‘They ALL have been here twice.’ b. Tamen dou lai -guo [LIANGF -ci] -le. they DOU come -EXP two-time -ASP ‘They’ve been here twice.’ Being here twice is a lot. c. (Lian) [TAMEN]F dou lai -guo liang-ci -le. LIAN they DOU come -EXP two-time -ASP ‘Even THEY have been here twice.’

The goal of this paper is to provide a uniform semantics of dou to ac- count for its seemingly diverse functions. I propose that dou is a special ex- The Mandarin Particle dou: A Pre-exhaustification Exhaustifier 5 haustifier that operates on sub-alternatives and has a pre-exhaustification effect. The basic idea can be roughly described as follows. Assume that a dou-sentence is of the form “dou(x, P)” where x and P correspond to the associated item and the remnant VP, respectively. The basic meaning of dou(x, P) is P(x) and not only P(x 0), where x 0 can be a proper subpart of x, a weaker scale-mate of x, and so on. For example, “[A and B] dou came” means that A and B came, not only A came, and not only B came; “it’s dou [five] o’clock” means that it’s 5 o’clock, not just 4, not just 3, .... The rest of this paper is organized as follows. Section 2 will review two representative theories on the semantics of dou, namely the distributor approach (Lin 1998) and the maximality operator approach (Giannaki- dou & Cheng 2006; Xiang 2008). Section 3 will define dou as a special exhaustifier and compare it with the canonical exhaustifier only. Section 4 will discuss the universal quantifier use of dou. I will show that the so called “distributivity requirement” and “plurality requirement” are both illusions, and that the facts that usually thought to be related to these two requirements result from the additive presupposition of dou. Section 5 and 6 will be centered on the FCI-licenser use and the scalar marker use, respectively.

2 Previous studies There are numerous studies on the syntax and semantics of dou. Earlier approaches treat dou as an adverb with universal quantiﬁcation power (Lee 1986; Cheng 1995; a.o.). Portner (2002) analyzes the scalar marker use of dou in a way similar to the inherent scalar semantics of the En- glish focus sensitive particle even. Hole (2004) treats dou as a universal quantiﬁer over the domain of alternatives. This section will review two representative studies on the semantics of dou, one is the distributor approach by Lin (1996), and the other is the maximality operator approach along the lines of Giannakidou & Cheng (2006) and Xiang (2008).

2.1 The distributor approach Lin (1996, 1998) provides the ﬁrst extensive treatment of the semantics of dou. He proposes that dou is an overt counterpart of the generalized distributor D in the sense of Schwarzschild (1996). Unlike the regular distributor each which distributes over an atomic domain, the generalized 6 Y. Xiang

D-operator distributes over the cover of the nominal phrase associated with dou. A cover of an individual x is a set of subparts of x, as deﬁned in (11) and exempliﬁed in (12). Its value is determined by the linguistic and non-linguistic context.

(11) Cov is a cover of x iff (i) Cov is a set of subparts of x; and (ii) every subpart of x belongs to some member in Cov.

(12) Possible covers of abc and corresponding readings:

a, b, c (atomic distributive)  { }   a b, c  a{ ⊕b, b }c (nonatomic distributive)  { ⊕ ... ⊕ }  a b c (collective) { ⊕ ⊕ } The semantics of dou is thus schematized as follows.

(13) ¹douº(P, x) is true iff x D(Cov)(P) iff y∈ Cov[P(y) = 1], where Cov is a cover of x. ∀ ∈ The distributor approach only considers the quantifier use of dou. It is unclear how one can extend it to the other uses such as the FCI-licenser use and the scalar marker use. Moreover, even for the quantifier use, this approach faces the following challenges. First, dou evokes a distributivity requirement, but the generalized D- distributor does not. For instance, as seen in (3b) and repeated below, the presence of dou eliminates the collective reading of the prejacent sentence. As Xiang (2008) argues, if dou were a generalized distributor, it should be compatible with a single cover reading (viz. the collective reading): there can be a discourse under which the cover of tamen ‘they’ denotes a singleton set like a b c ; distributing over this singleton set yields a collective reading.{{ ⊕ ⊕ }} The Mandarin Particle dou: A Pre-exhaustification Exhaustifier 7

(14) [Tamen] dou mai -le fangzi. they DOU buy -PERF house ‘They dou bought houses.’ (# collective)

Second, unlike English distributors like each and all1, dou can be associated with a distributive expression such as NP-gezi ‘NP each’.2

(15) a. They each (*each/*all) has some advantages. b. [Tamen gezi] dou you yixie youdian. They each DOU have some advantage ‘They each dou has some advantages .’

2.2 The maximality operator analysis Another popular approach, initiated by Giannakidou & Cheng (2006) and extended by Xiang (2008), is to treat dou as a presuppositional maximality operator. Brieﬂy speaking, this approach proposes that dou operates on a non-singleton cover of the associated item, returns the maximal plural element in this cover, and presupposes the existence of this maximal plural element. I schematize this idea as follows.

(16) Let Cov be a cover of x, then ¹douº(x) = Cov > 1 y Cov[ Atom(y) z Cov[z y]]. | | ∧ ∃ι y ∈ Cov[¬Atom(y) ∧ ∀z ∈ Cov[z ≤ y]]. ∈ ¬ ∧ ∀ ∈ ≤ (¹douº(x) is deﬁned only if the cover of x is non-singleton and has a unique non-atomic maximal element; when deﬁned, the

1Champollion (2015) argues that all is a distributor that distributes down to sub- groups, while that each distributes all the way down to atoms. 2Similar arguments have been reached in previous studies (Cheng 2009; a.o.), but they are mostly based on the fact that dou can be associated with the distributive quantificational phrase mei-CL-NP ‘every NP’, as exemplified in (i). This fact, however, cannot knock down the distributor approach for the quantifier use of dou: observe in (i) that stress falls on the distributive phrase mei-CL-NP,not the particle dou; therefore here dou functions as a scalar marker, rather than a quantifier.

(i) a. [MEI-ge ren] dou you youdian. every-CL person DOU have advantage ‘Everyone dou has some advantages.’ b.?? [Mei-ge ren] DOU you youdian. every-CL person DOU have advantage 8 Y. Xiang

reference of ¹douº(x) is this maximal element.) This approach is close to the standard treatment of the definite deter- miner the (Sharvy 1980; Link 1983): the picks out the unique maximal element in the extension of its NP complement and presupposes the existence of this maximal element. This approach is superior to the distributor approach in two respects: first, it captures the maximality requirement; and second, it can be extended to the scalar use of dou (see details in Xiang 2008). Nevertheless, this approach still faces several conceptual or empirical problems. First, the plurality requirement comes as a stipulation on the presupposition of dou: dou presupposes that the selected maximal element is non-atomic. It is unclear why is that so; the definite article the does not trigger such a plural presupposition. Moreover, as we will see in section 4.3, this plural presupposition is neither sufficient nor necessary in deal- ing with the relevant facts. Second, this approach predicts no distributivity effect at all. Under this approach, “[X ] dou did f ” only asserts that the maximal element in the cover of X did f , not that each element in the cover of X did f . For instance in (14), if the cover of tamen ‘they’ is a b, a b c , the predicted assertion is simply that a b c bought{ houses⊕ , which⊕ ⊕ says} nothing as to whether a b bought houses.⊕ ⊕ ⊕ 3 Defining dou as a special exhaustifier This section will start with the semantics of the canonical exhaustifier only, and then define Mandarin particle dou as a special exhaustifier: dou is a pre-exhaustification exhaustifier that operates on sub-alternatives.

3.1 Canonical exhaustifier only The exclusive particle only is a canonical exhaustifier. Using Alternative Semantics for focus (Rooth 1985, 1992, 1996), we can summarize the standard treatment of the semantics of only into two parts. First, a focused element is associated with a set of focus alternatives. This alternative set grows point-wise (Hamblin 1973), as recursively defined in (17), adopted from Chierchia (2013: 138).

(17) a. Basic Clause: for any lexical entry α, Alt(α) = The Mandarin Particle dou: A Pre-exhaustiﬁcation Exhaustiﬁer 9

(i) {¹αº} if α is lexical and does not belong to a scale; (ii) {¹α1º, ..., ¹αnº} if α is lexical and part of a scale ¹α1º, 〈 ..., ¹αnº . b. Recursive Clause:〉 Alt(β(α)) = b(a) : b Alt(β), a Alt(α) { ∈ ∈ } Second, the exclusive particle only presupposes the truth of its prejacent proposition (Horn 1969) and asserts an exhaustivity inference. This exhaustivity inference negates all the focus alternatives that are excludable. An alternative is excludable as long as it is not entailed by the prejacent.3

(18) a. ¹onlyº(p) = p. q Excl(p)[ q] (To be revised) b. Excl(p) = q : q∀ Alt∈ (p) p ¬ q { ∈ ∧ 6⊆ } In addition to the prejacent presupposition, I argue that only also trig- gers an additive presupposition, namely that the prejacent has at least one excludable alternative. In (19), only has a restricted exhaustiﬁcation domain, namely {I will invite John, I will invite Mary, I will invite John and Mary}. Contrary to the case of (19a), (19b) is infelicitous because the prejacent I will invite both John and Mary is the strongest one among the alternatives and has no excludable alternative. As Martin Hackl (p.c.) points out, the additive presupposition of only can be reduced to a more general economy condition that an overt operator cannot be applied vac- uously. For sake of comparison, observe that (19c) is felicitous, which is because covert exhaustiﬁcation is free from the economy condition and does not trigger an additive presupposition.

(19) Which of John and Mary will you invite?

a. Only JOHNF , (not Mary/ not both). b. # Only BOTHF . c. BOTHF . In sum, I schematize the semantics of only as follows: it presupposes the truth of its prejacent and the existence of an excludable alternative; it negates each excludable alternative.

3For simplicity, here and throughout the paper, propositional letters like p are sloppily used for both syntactic expressions and semantic values. 10 Y. Xiang

(20) ¹onlyº(p) = p q Excl(p). q Excl(p)[ q] (Final version) a. Prejacent∧ presupposition: ∃ ∈ ∀p ∈ ¬ b. Additive presupposition: q Excl(p) c. Assertion: q Excl(p)[ ∃q] ∈ ∀ ∈ ¬ 3.2 Special exhaustifier dou I define dou as a pre-exhaustification exhaustifier over sub-alternatives, as schematized in (21): it presupposes an additive inference; it affirms the prejacent and negates the exhaustification of each sub-alternative.

(21) ¹douº(p) = q Sub(p).p q Sub(p)[ O(q)] ∃ ∈ ∧ ∀ ∈ ¬ The additive presupposition is motivated by the economy condition, just like what we saw with the canonical exhaustiﬁer only. The exhaustivity inference asserted by dou differs from that asserted by only in two respects. First, only operates on excludable alternatives, but dou operates on sub-alternatives. For now we can understand sub-alternatives as weaker alternatives, or say, the alternatives that are not excludable and distinct from the prejacent. A revision will be made in section 5.

(22) Sub(p) = q : q Alt(p) p q (To be revised) = Alt{ (p)∈ Excl(p∧) ⊂p } − − { } Second, dou has a pre-exhaustification effect: it negates the “exhaustification” of each sub-alternative. The pre-exhaustification effect is realized by applying an O-operator to each sub-alternative.4 The O-operator is a covert counterpart of the exclusive particle only, coined by the grammatical view of scalar implicatures (Fox 2007; Chierchia et al. 2013; Fox & Spector to appear; a.o.). This O-operator affirms the prejacent and negates all the excludable alternatives of the prejacent.

(23) O(p) = p q Excl(p)[ q] (Chierchia et al. 2013) ∧ ∀ ∈ ¬ Consider (24) for a simple illustration of the present deﬁnition. The prejacent and the sub-alternatives are schematized in (24a) and (24b),

4In section 6, we will see other options to derive the pre-exhaustification effect. For instance, when dou is used as a scalar marker, the pre-exhaustification effect is realized by applying a scalar exhaustifier ( just) to the sub-alternatives. ≈ The Mandarin Particle dou: A Pre-exhaustification Exhaustifier 11 respectively. Exercising dou affirms the prejacent and negates the exhaustification of each sub-alternative, as in (24c), yielding the inference that John and Mary arrived, not only John arrived, and not only Mary arrived. The exhaustivity inference (underlined above) is entailed by the prejacent and attributes nothing new to the truth conditions.5

(24) [John and Mary] dou arrived. a. p = A(j m) b. Sub(p) =⊕ A(j), A(m) { } c. ¹douº(p) = A(j m) O[A(j)] O[A(m)] ⊕ ∧ ¬ ∧ ¬ 4 The universal quantiﬁer use Recall that dou evokes three requirements when used as a universal quan- tiﬁer: (i) the “maximality requirement”, namely that dou forces maximality with respect to the domain denoted by the associated item; (ii) the “distributivity requirement”, namely that the prejacent sentence cannot take a collective reading; (iii) the “plurality requirement”, namely that the item associated with dou must take a non-atomic interpretation. In this section, I will show that the maximality requirement is a simple logical consequence of dou’s assertion, and argue that the other two requirements are illusions. Moreover, I will argue that all the facts that are thought to result from the latter two requirements actually result from the additive presupposition of dou.

4.1 Explaining the “maximality requirement” The maximality requirement comes from the assertion of dou: dou asserts that, for each sub-alternative of the prejacent, its exhaustiﬁcation is false.

5One might wonder why dou is used if it does not change the truth conditions. Such uses are observed cross-linguistically. For instance in (i), the distributor both changes nothing to the truth conditions. One possibility, as raised by the audience at LAGB 2015, is that dou and both are used for the sake of contrasting with non-maximal operators like only part of or only one of. If this is the case, the question under discussion for (24) and (i) would be as follows: it is the case that John and Mary both arrived or that only one of them arrived?

(i) John and Mary BOTH arrived. 12 Y. Xiang

For instance in (2), repeated below, applying dou yields the inference that it is false that only a subgroup of children went to the park; therefore in presence of dou, (25) is true iff all the children went to the park.

(25) [Haizimen] (#dou) qu -le gongyuan. children DOU go -PERF park ‘The children (#all) went to the park.’

4.2 Explaining the “distributivity requirement” To generate sub-alternatives and satisfy the additive presupposition of dou, the prejacent of dou needs to be monotonic with respect to the position associated with dou, which therefore gives rise to the “distributivity requirement”. For instance, the sentence (26) rejects a collective reading because under this reading the prejacent proposition of dou is non- monotonic with respect to the subject position and hence has no sub- alternative, as shown in (26a). In contrast, when taking an atomic or a non-atomic distributive reading, the prejacent of dou is monotonic with respect to the subject position and does generate some sub-alternatives, as shown in (26b) and (26c).6

(26) [abc] dou bought houses. a. Collective (i) abc together× BH. ab together BH. (ii) Sub (abc together6⇒ BH)= ; b. Atomic distributive p (i) abc each BH. ab each BH. (ii) Sub (each(x)(⇒BH)) = {each(x)(BH): x < abc } c. Nonatomic distributive p (i) members of Covabc each BH. members of X each BH, where X Covabc ⇒ ⊂ (ii) Sub (D(Covabc)(BH)) = D(X )(BH) : X Covabc { ⊂ } Hence, dou itself is not a distributor; but in certain cases, the additive presupposition of dou evokes the use of a distributor (a covert each or a covert generalized distributor). We can now easily explain why dou can be

6 Covabc in (26c) stands for a contextually determined variable that is a cover of abc. The Mandarin Particle dou: A Pre-exhaustiﬁcation Exhaustiﬁer 13 associated with a distributive expression NP-gezi ‘NP-each’: the presence of the distributor gezi ‘each’ is actually required for the sake of satisfying the additive presupposition of dou; if gezi is not overtly used, a covert distributor is still present in the logical form.

(27) [Tamen gezi] dou you yixie youdian. They each DOU have some advantage ‘They each dou has some advantages .’

Moreover, dou can be applied to a collective statement as long as this statement satisﬁes the monotonicity requirement. For instance, dou is compatible with monotonic collective predicates (e.g., shi pengyou ‘be friends’, jihe ‘gather’, jianmian ‘meet’), as shown in (28). Consider (28a) for instance. Let tamen ‘they’ denote three individuals abc. The set of sub- alternative sets is {ab are friends, bc are friends, ac are friends}; applying dou yields the following inference: abc are friends, not only ab are friends, not only bc are friends, and not only ac are friends.

(28) a. [Tamen] (dou) shi pengyou. they DOU be friends ‘They are (all) friends.’ b. [Tamen] (dou) zai dating jihe -le. they DOU at hallway gather -ASP ‘They (all) gathered in the hallway.’ c. [Tamen] (dou) jian-guo-mian -le. they DOU see-EXP-face -ASP ‘They (all) have met.’

By comparison, dou cannot be applied to a collective statement that does not satisfy the monotonicity requirement, as shown in (29).

(29) [Tamen] (*dou) zucheng -le lia er-ren-zu. they DOU form -ASP two two-person-group ‘They (*all) formed two pairs.’

Note to distinguish the case in (29) from the following ones, where the prejacent sentences actually admit non-collective (viz., non-atomic distributive) readings and thus satisfy the monotonicity requirement. 14 Y. Xiang

(30) [Tamen] dou zucheng -le er-ren-zu. they DOU form -ASP two-person-group ‘They all formed pairs.’ (31) [Women he tamen] dou zucheng -le lia er-ren-zu. we and they DOU form -ASP two two-person-group ‘We formed two pairs, and they formed two pairs.’

In (30), the extension of the predicate ‘formed pairs’ (FP) is closed under sum, just like any plural terms: FP(a b) FP(c d) FP(a b c d) (see Kratzer (2008) for general discussions⊕ on∧ pluralizing⊕ ⇒ verbal⊕ predicates);⊕ ⊕ hence the prejacent sentence admits a covered/cumulative reading. In (31), although the predicate ‘formed two pairs’ (F2P) is non-monotonic, the subject ‘we and they’ can be interpreted as a higher-order conjunction, each conjunct of which yields a sub-alternative. A schematized derivation for the sub-alternatives in (31) is given in (32).

(32) a. ¹we and theyº = λPest .λw.Pw(we) Pw(they) ∧ b. ¹we and they F2Pº = λw.F2Pw(we) F2Pw(they) c. Sub(we and they F2P) = {F2P(we), ∧F2P(they)}

4.3 Explaining the “plurality requirement” I argue that the so-called “plurality requirement ” of dou is illusive, and that the related facts all result from the additive presupposition of dou. First, the plurality requirement is unnecessary: dou can be associated with an atomic as long as the remnant VP denotes a divisive predicate.

(33) P is divisive iff x[P(x) y < x[P(y)]] (A predicate is divisive∀ iff→ whenever ∀ it holds of something, it also holds of each of its proper parts.

For instance in (34a), the associated item ‘that apple’ takes only an atomic interpretation; with a divisive predicate λx. John ate x, the prejacent sentence of dou has sub-alternatives, as schematized in (35a), which supports the presupposition of dou. In contrast, in (34b), the predicate λx. John ate half of x is not divisive and hence is incompatible with the use of dou. The Mandarin Particle dou: A Pre-exhaustiﬁcation Exhaustiﬁer 15

(34) a. Yuehan ba [na-ge pingguo] (dou) chi -le. John BA that-CL apple DOU eat -PERF ‘John ate that apple.’ b. Yuehan ba [na-ge pingguo] (*dou) chi -le yi-ban. John BA that-CL apple DOU eat -PERF one-half Intended: ‘John ate half of that apple.’ (35) a. ‘J ate that apple.’ ‘J ate x.’ (x < that apple) Sub (J ate that apple⇒ ) = {J had x: x < that apple} b. ‘J ate half of that apple.’ ‘J ate half of x.’ (x < that apple) Sub (John ate half of that6⇒ apple) = ; Second, the plurality requirement is insufﬁcient. When applied to a monotonic collective statement, dou requires its associated item to denote a group consisting of at least three members, as exempliﬁed in (36).

(36) [Tamen -sa/*-lia] dou shi pengyou. they -three/-two DOU be friends ‘They three/*two are all friends.’ This fact is also predicted by the additive presupposition. As schematized in (37), the proper subparts of an dual-individual denotes an atomic individual, which however is undeﬁned for the collective predicate ‘be friends’. Hence, if the item associated with dou in (36) denotes only a dual-individual, the prejacent of dou has no sub-alternative, which therefore leaves the presupposition of dou unsatisﬁed.

(37) [ab] (*dou) are friends. a. ¹be friendsº = λx.Atom(x) = 0.be-friends(x) b. Sub(ab are friends) = ; 5 The universal FCI-licenser use Dou can license the universal FC use of polarity items, wh-items, and pre-verbal disjunctions. In this section, I argue that the asserted compo- nent of dou converts a disjunctive/existential statement into a conjunc- tive/universal statement, giving rise to an FC inference. I will also explain why the licensing of universal FCIs requires the presence of dou, and why the licensing of a pre-verbal disjunction as a universal FCI exhibits the 16 Y. Xiang effect of modal obviation.

5.1 Licensing conditions of Mandarin FCIs In Mandarin, the licensing of a universal FCI requires the presence of dou. For instance in (38), the bare wh-word shei ‘who’ is licensed as a universal FCI only when it precedes dou.

(38) [Shei] *(dou) jiao -guo jichu hanyu. who DOU teach -EXP intro Chinese. ‘Everyone has taught Intro Chinese.’

To license the universal FC use of a disjunction, dou must be present and followed by a possibility modal, as shown in (39) and (40).

(39) [Yuehan huozhe Mali] dou keyi/*bixu jiao jichu hanyu. John or Mary DOU can/must teach intro Chinese ‘Both John and Mary can teach Intro Chinese.’ (40) [Yuehan huozhe Mali] (*dou) jiao -guo jichu hanyu. John or Mary DOU teach -EXP intro Chinese Intended: ‘Both Johan and Mary have taught Intro Chinese.’

This requirement is also observed with English emphatic item any: as shown in (41), any is licensed as a universal FCI when it precedes a possibility modal, but not licensed when it appears in an episodic statement or before a necessity modal.

(41) a. *Anyone came in. b. Anyone can/*must come in.

The licensing conditions of na-CL-NP ‘which-NP’ and renhe-NP ‘any- NP’ are less clear. Giannakidou & Cheng (2006) claim that the universal FC uses of these items are only licensed in a pre-dou+◊ position; their judgements are illustrated in (42). Nevertheless, it is difﬁcult to justice the data because judgements on (42) vary greatly among native speakers.

(42) a. [Na-ge/Renhe -ren] dou keyi/?bixu lai. which-CL/anywhat -person DOU can/must come Intended: ‘Everyone can/must come.’ The Mandarin Particle dou: A Pre-exhaustiﬁcation Exhaustiﬁer 17

b.? [Na-ge/Renhe -ren] dou lai -guo. which-CL/anywhat -person DOU come -ASP Intended: ‘Everyone has been here.’

Despite the noise in the judgments, the licensing conditions of universal FCIs in Mandarin can be summarized as follows. First, every universal FCI requires the presence of dou. Second, every universal FCI can be licensed before dou+◊. Third, in absence of the possibility modal, ‘which’/‘any’-NP is less likely to be licensed than bare wh-words, but more likely to be licensed than disjunctions. For other recent studies, see Liao (2011), Cheng & Giannakidou (2013), and Chierchia & Liao (2015).

5.2 Predicting the universal FC inferences Wh-items are generally considered as existential indefinites; thus in (38), repeated below, the prejacent sentence of dou is a disjunction, and the sub-alternatives are the disjuncts. Applying dou affirms the prejacent and negates the exhaustification of each disjunct, yielding a universal FC inference. In a word, dou turns a disjunction into a conjunction.

(43) [Shei] *(dou) has taught Intro Chinese. a. p = f (a) f (b) b. Sub(p) =∨ f (a), f (b) { } c. ¹douº(p) = [f (a) f (b)] O f (a) O f (b) = [f (a) ∨ f (b)] ∧[ ¬f (a) ∧f ¬(b)] [f (b) f (a)] = [f (a) ∨ f (b)] ∧ [f (a) → f (b)]∧ → = f (a) ∨f (b) ∧ ↔ ∧ What makes the use of dou mandatory in (38)? Following Liao (2011) and Chierchia & Liao (2015), I assume that the sub-alternatives associated with a Mandarin wh-word are obligatorily activated in a non-interrogative sentence and must be used up via exercising a c-commanding exhaustifier.7 If dou is absent, these sub-alternatives would be used by a basic exhaustifier (23), repeated in (44a), which has no pre-exhaustification effect. As schematized in (44b), a basic O-operator affirms the prejacent

7In the case of disjunctions, sub-alternatives are simply what usually call “domain alternatives”, evoked by domain widening. (Krifka 1995; Lahiri 1998; Chierchia 2006) 18 Y. Xiang disjunction and negates both disjuncts, yielding a contradiction.8

(44) a. O(p) = p q Excl(p)[ q] (Chierchia et al. 2013) b. O(f (a) ∧f ( ∀b))∈ = [f (a) ¬f (b)] f (a) f (b) = ∨ ∨ ∧ ¬ ∧ ¬ ⊥ Now, a problem arises as to the definition of sub-alternatives: in section 3, I defined sub-alternatives as weaker alternatives, namely alternatives that are not excludable and distinct from the prejacent; but in (43) the disjuncts are semantically stronger than the disjunction. This problem can be solved by a simple move from excludability to innocent excludability, a notion proposed by Fox (2007): an alternative is innocently excludable iff the inference of affirming the prejacent and negating this alternative is consistent with negating any excludable alternative. Thus, we can say that sub-alternatives are alternatives that are not innocently excludable and distinct from the prejacent.

(45) a. Excludable alternatives Excl(p) = q : q Alt(p) p q (the set of{ alternatives∈ that∧ are6⊆ entailed} by the prejacent.) b. Innocently excludable alternatives (Fox 2007)

IExcl(p) = q : q Alt(p) q0 Excl(p)[p q q0] ({q: afﬁrming{ p ∈and negating∧ ¬∃ q does∈ not entail∧ ¬ any→ exclud-} able alternatives}) c. Sub-alternatives Sub(p) = Alt(p) IExcl(p) p (ﬁnal version) (the set of alternatives− excluding− { } the innocently excludable alternatives and the prejacent)

In (43), the disjuncts are not innocently excludable to the disjunction: as schematized below, afﬁrming the disjunction and negating one of the disjuncts entail the other disjunct; in other words, afﬁrming the disjunction and negating both disjuncts would yield a contradiction. Hence, the sub-alternatives of a disjunction are the disjuncts.

(46) [[f (a) f (b)] f (a)] f (b) ∨ ∧ ¬ ⇒ 8Basic disjunctions are free from this problem, because they do not mandatorily evoke sub-alternatives. See Chierchia (2006) for discussions on activations of alternatives. The Mandarin Particle dou: A Pre-exhaustiﬁcation Exhaustiﬁer 19

A full deﬁnition of dou is schematized as follows.

(47) a. ¹douº(p) = q Sub(p).p q Sub(p)[ O(q)] (i) Presupposition:∃ ∈ p has∧ some ∀ ∈ sub-alternatives.¬ (ii) Assertion: p is true, while the exhaustiﬁcation of each sub-alternative of p is false. b. Sub(p) = Alt(p) IExcl(p) p (the set of alternatives− excluding− { } the innocently excludable alternatives and the prejacent)

Readers who are familiar with the grammatical view of exhaustiﬁca- tions might ﬁnd that dou is similar to the operation of recursive exhausti-

fications OR proposed by Fox (2007). Using the notations in (47), I define Fox’s recursive exhaustifier OR as follows.

(48) OR(p) = p q Sub(p)[ O(p)] q0 IExcl(p)[ q0] ∧ ∀ ∈ ¬ ∧ ∀ ∈ ¬ Thus dou is weaker than OR: dou does not negate the innocently excludable alternatives; therefore, applying dou to a disjunction does not generate an exclusive inference. For instance, the sentence (39) does not imply the exclusive inference that only John and Mary can teach Intro Chinese.

5.3 Modal Obviation Recall the contrast between disjunctions and bare wh-words: dou alone is sufﬁcient to license the universal FC use of a bare wh-word, but not that of a disjunction. To capture this contrast, I assume that disjunctions evoke scalar implicatures, while bare wh-words do not (compare Liao 2011; Chierchia & Liao 2015). Compare the episodic sentences in (49). In both examples, exercising dou yields an FC inference that John&Mary/everyone have/has taught Intro Chinese; but (49a) also evokes a scalar implicature, namely that not both John and Mary have taught Intro Chinese.

(49) a. [Yuehan huozhe Mali] (*dou) jiao -guo jichu hanyu. John or Mary DOU teach -EXP intro Chinese b. [Shei] *(dou) jiao -guo jichu hanyu. who DOU teach -EXP intro Chinese ‘Everyone has taught Intro Chinese.’ 20 Y. Xiang

Hence, dou cannot be used in (49a) because it yields a universal FC inference which contradicts the scalar implicature, à la Chierchia’s (2013) explanation on the licensing condition of the FC any. In absence of dou, the sub-alternatives of a disjunction are not activated, and then (49a) simply means that John or Mary but not both has taught Intro Chinese. A pre-verbal disjunction is licensed as a universal FCI when it appears before dou+◊. This effect is called “modal obviation”, namely that the presence of a possibility modal eliminates the ungrammaticality. This effect is also observed with the English any, as seen in (41).

(50) a. [Yuehan huozhe Mali] dou keyi jiao jichu hanyu. John or Mary DOU can teach intro Chinese ‘Both John and Mary can teach Intro Chinese.’ b. [Yuehan huozhe Mali] (*dou) bixu jiao jichu hanyu. John or Mary DOU must teach intro Chinese ‘Both John and Mary must teach Intro Chinese.’

There have been plenty of discussions on the phenomenon of Modal Ob- viation involved in licensing universal FCIs. Representative works include Dayal (1998, 2013), Giannakidou (2001), Chierchia (2013), and among the others. This paper is not in a position to do full justice to these discussions, but just adds one more accessible story to the market. I propose that the scalar implicature of a pre-verbal disjunction can be assessed within a circumstantial modal base: the modal base is restricted to the set of worlds where the scalar implicature is satisﬁed. For instance, (50) intuitively suggests that the speaker is only interested in cases where exactly one person teaches Intro Chinese. Assume that the property teach Intro Chinese denotes only three world-individual pairs, as in (51a). For instance the pair w1, j is read as only John teaches Intro Chinese in w1. The scalar implicature〈 { of}〉 the pre-verbal disjunction restricts the modal base into M, namely the set of worlds where not both John and Mary teach Intro Chinese. Exercising dou yields the universal FC inferences in (51c) and (51d). Crucially, only (51c) is true under M.

(51) a. f = w1, j , w2, m , w3, j, m b. M ={〈w1, w{ 2}〉 〈 { }〉 〈 { }〉} { } c. ¹douº [◊f (j) ◊f (m)] = ◊f (j) ◊f (m) True under M ∨ ∧ The Mandarin Particle dou: A Pre-exhaustiﬁcation Exhaustiﬁer 21

d. ¹douº [f (j) f (m)] = f (j) f (m) False under M ∨ ∧ More broadly speaking, there is no modal base, except the empty one, under which (51d) is true; therefore necessity modals cannot obviate the contradiction between the FC inference and the scalar implicature. If I am on the right track, as for the licensing conditions for the universal FC uses of na-CL-NP and renhe-NP,whether a speaker accepts (42) in absence of the possibility modal is determined by whether he interprets these items with scalar implicatures.

6 Scalar marker When dou is associated with a scalar item or occurs in the focus construction [lian Foc dou ...], it functions as a scalar marker. In such a case, sub-alternatives are the alternatives ranked lower than the prejacent with respect to a contextually relevant probability measure, and the pre-exhaustification effect is realized by the scalar exhaustifier JUST. In the following, I will firstly sketch out the semantics of a scalar dou, and then capture the ‘even’-like interpretation and the licensing conditions of minimizers in the [lian Foc/Min dou ...] construction. 6.1 Association with a scalar item When dou is associated with a scalar item, the sub-alternatives are alternatives that rank lower than the prejacent proposition in the considered scale, as schematized in (52), where q <µ p says that q is less likely than p with respect to some contextually relevant probability measure µ. For instance, in (53), sub-alternatives are propositions that rank lower than the prejacent in chronological order.

(52) Sub(p) = q : q C q <µ p ({q: q is a{ contextually∈ ∧ relevant} alternative of p that rank lower than p in the considered scale}) (53) Dou [WU-dian] -le. DOU ﬁve-o’clock -ASP ‘It is dou [FIVE] o’clock.’ a. Sub(it’s 5 o’clock ) = {it’s 4 o’clock, it’s 3 o’clock, ...} b. ¹dou[it’s 5 o’clock]º = ‘it’s 5, not just 4, not just 3, ...’ 22 Y. Xiang

Since here the alternatives are ordered based on their strength in the considered scale, the pre-exhaustification effect of dou is realized by the scalar exhaustifier JUST. Therefore, dou is defined as in (54) when it functions as a scalar marker.

(54) ¹douº(p) = q Sub(p).p q Sub(p)[ JUST(p)] ∃ ∈ ∧ ∀ ∈ ¬ a. Sub(p) = q : q C q <µ p (the set of{ contextually∈ ∧ relevant} alternatives that are ranked lower than p in the considered scale) b. JUST(p) = λw.p(w) q C[q(w) q µ p] (p is true; every contextually∧ ∀ ∈ relevant→ true≤ alternative of p does not rank higher than p in the considered scale)

To generate sub-alternatives and satisfy the additive presupposition of dou, the prejacent needs to be relatively strong among the quantiﬁcational statements. For instance in (55), dou can be associated with ‘many-NP’ but not with ‘few-NP’.

(55) [Duo/*Shao -shu -ren] dou lai -le. many/less -amount -person DOU come -ASP ‘Most/*few people dou came.’

6.2 The [lian Foc dou...] construction In a [lian Foc dou ...] construction, alternatives are ordered with respect to likelihood. Sub-alternatives are focus alternatives that are less unlikely (i.e., more likely) to be true than the prejacent, as schematized in (56). Here the variable C is a set of focus alternatives that are contextually relevant. This deﬁnition is a natural transition from informativity to likelihood: a proposition that is less informative is less unlikely to be true.

(56) Sub(p) = q : q C q

(57) Lian [duizhang]F dou chidao -le. LIAN team-leader DOU late -ASP ‘Even the team leader was late.’

Extending the deﬁnition of dou to the [lian Foc dou ...] construction, I schematize the meaning of dou as follows. The underlined inference, namely that every alternative that is less unlikely to be true than p is less unlikely to be true than some true alternative, is asymmetrically entailed by the rest asserted part, namely that p is true. Hence, the asserted com- ponent of dou simply afﬁrms its prejacent.

(58) ¹douº(p) = q Sub(p).p(w) q Sub(p)[ JUST(p)(w)] ∃ ∈ ∧ ∀ ∈ ¬ = λw. q C[q

9Note that this additive presupposition says nothing about the truth value of any sub-alternative, as shown in (i).

(i) Lian [Yuehan] dou jige -le, qita-ren zenme mei -you? LIAN John DOU pass -ASP, other-person how NEG -ASP. ‘Even John passed [the exam]. Why is that the others didn’t?’ 24 Y. Xiang

6.3 Association with a minimizer Observe that, in licensing a minimizer, the post-dou negation is mandatory in (59a) but optional in (59b).

(59) a. Yuehan (lian) [YI-ge ren]F *(dou) *(bu) renshi. John LIAN one-CL person DOU NEG know ‘John doesn’t know anyone.’ b. Yuehan (lian) [YI-fen qian]F *(dou) (bu) yao. John LIAN one-cent money DOU NEG request Without negation: ‘John even doesn’t want one cent.’ With negation: ‘John wants it even if it is just one cent.’

I argue that the distributional pattern of the post-dou negation in a [lian MIN dou (NEG) ... ] construction is also constrained by the additive presupposition of dou. The additive presupposition of dou requires the prejacent not to be weakest proposition among the alternatives. In (59a), this requirement forces the minimizer ‘one person’ to take reconstruction and gets interpreted below negation: there is at least one person that John didn’t invite is weaker than there are at least N people that John didn’t invite where N > 1; while [John invited at least one person] is stronger than any propositions of¬ the form [John invited at least N people] where N > 1. Compare, (60) is ungrammatical:¬ the minimizer ‘one person’ is a subject and its reconstructed position is higher than negation; hence there is no structure under which the presupposition of dou is satisﬁed.

(60) * (Lian) [YI-ge ren]F dou bu renshi Yuehan. LIAN one-CL person DOU NEG know John.

In (59b), under the assumption that John shouldn’t want the money if the money is too little, we expect that John wants one cent is more unlikely to be true than John wants two cents; therefore, the additive presupposition of dou can be satisﬁed even in absence of the post-dou negation.

7 Conclusions In this paper, I offered a uniform semantics to capture the seemingly diverse functions of Mandarin particle dou, including the quantifier use, the The Mandarin Particle dou: A Pre-exhaustification Exhaustifier 25

FCI-licenser use, and the scalar use. I proposed that dou is a special exhaustifier that operates on sub-alternatives and has a pre-exhaustification effect: dou presupposes the existence of at least one sub-alternative, asserts the truth of the prejacent and the negation of each pre-exhaustified sub-alternative. In a basic case, sub-alternatives are alternatives that are not innocently excludable and distinct from the prejacent, and the pre-exhaustification effect is realized by a regular exhaustifier (viz. the O-operator). Depend- ing on the meaning of its associated item, dou functions either as a universal quantifier or as a universal FCI-licenser. When dou is applied to a scalar statement, sub-alternatives are alternatives less unlikely than the prejacent sentence with respect to the considered probability measure, and the pre-exhaustification effect is realized by the scalar exhaustifier JUST. The additive presupposition of dou explains the distributional pattern of dou and many of its semantic consequences, such as the requirements regarding to distributivity, plurality, and monotonicity, the ‘even’-like interpretation of [lian Foc dou ...] construction, the distributional pattern of the post-dou negation in licensing minimizers, and so on.

Acknowledgments I thank Gennaro Chierchia, Dany Fox, Daniel Hole, C.-T. James Huang, Mingming Liu, Edwin C.-Y.Tsai, Ming Xiang, the reviewers of CSSP 2015, and the audience at LAGB 2015 and EACL 9 for helpful comments and discussions. All errors are mine.

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